The keypoint methodology has a number of advantages, such as robustness to occlusion or missing correspondences, invariance to translation, scaling, rotation and intensity contrast variations between images. Our work adopts the 3D SIFT-Rank [1]} representation, a robust, general tool used for a variety of imaging tasks, including registration [1]}, [3]}, whole body segmentation [4]}, classification [5]}, [6]}, regression [7]}, without the need for context-specific training procedures or data. As in the original 2D SIFT approach, keypoint geometry is represented as a location \(\bar{x}\)  and scale \(\sigma \) , these are detected as extrema of a difference-of-Gaussian scale-space [8]}, as \(\operatornamewithlimits{argmax}_{\bar{x},\sigma } \ |I(\bar{x},\sigma ) \, - \, I(\bar{x},\kappa \sigma )|\) , approximating the Laplacian-of-Gaussian [9]}, where \(I(\bar{x},\sigma )=I(\bar{x})*\mathcal {N}(\bar{x},\sigma ^2)\)  represents the image convolved with the Gaussian kernel \(\mathcal {N}(\bar{x},\sigma ^2)\) . The geometry of a keypoint may thus be characterised as an isotropic Gaussian density centered on \(\bar{x}\)  with variance \(\sigma ^2\)  representing spatial extent \(\sigma \)  in 3D Euclidean space. Rotationally symmetric Gaussian filtering and uniformly sampled derivative operators lead to scale and rotation invariance. The 3D SIFT-Rank descriptor is a 64-element histogram of rank-ordered oriented gradients (HOG) [1]}, sampled about a scale-normalized reference frame centered on \(\bar{x}\)  and quantized uniformly into 2x2x2=8 spatial locations x 8 orientations. Rank transformation provides invariance to arbitrary monotonic variations of descriptor element values [11]}.
